Showing papers in "Nonlinearity in 2014"

Ground states for fractional Schrödinger equations with critical growth

Abstract: In this paper we study the existence and multiplicity of solutions for the critical fractional Schrodinger equation where e and λ are positive parameters, 0 2α, is the fractional critical exponent; V is a positive continuous potential satisfying some conditions and f is a continuous subcritical nonlinear term. We prove that the equation has a nonnegative ground state solution and investigate the relation between the number of solutions and the topology of the set where V attains its minimum, for all sufficiently large λ and small e.

Higher order averaging theory for finding periodic solutions via Brouwer degree

Abstract: In this paper we deal with nonlinear differential systems of the form where for i = 0, 1, ..., k, and are continuous functions, and T-periodic in the first variable, D being an open subset of , and e a small parameter. For such differential systems, which do not need to be of class , under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in e. Some applications are also performed.

Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability

Abstract: We describe a general N-solitonic solution of the focusing nonlinear Schrodinger equation in the presence of a condensate by using the dressing method. We give the explicit form of one- and two-solitonic solutions and study them in detail as well as solitonic atoms and degenerate solutions. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at a certain moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.

The real butterfly effect

Abstract: Historical evidence is reviewed to show that what Ed Lorenz meant by the iconic phrase 'the butterfly effect' is not at all captured by the notion of sensitive dependence on initial conditions in low-order chaos. Rather, as presented in his 1969 Tellus paper, Lorenz intended the phrase to describe the existence of an absolute finite-time predicability barrier in certain multi-scale fluid systems, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times. To distinguish from 'mere' sensitive dependence, the effect discussed in Lorenz's Tellus paper is referred to as 'the real butterfly effect'. Theoretical evidence for such a predictability barrier in a fluid described by the three-dimensional Navier–Stokes equations is discussed. Whilst it is still an open question whether the Navier–Stokes equation has this property, evidence from both idealized atmospheric simulators and analysis of operational weather forecasts suggests that the real butterfly effect exists in an asymptotic sense, i.e. for initial-time atmospheric perturbations that are small in scale and amplitude compared with (weather) scales of interest, but still large in scale and amplitude compared with variability in the viscous subrange. Despite this, the real butterfly effect is an intermittent phenomenon in the atmosphere, and its presence can be signalled a priori, and hence mitigated, by ensemble forecast methods.

Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time

Abstract: The ensemble Kalman filter (EnKF) is a method for combining a dynamical model with data in a sequential fashion. Despite its widespread use, there has been little analysis of its theoretical properties. Many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The aim of this paper is to initiate the development of a systematic analysis of the EnKF, in particular to do so for small ensemble size. The perspective is to view the method as a state estimator, and not as an algorithm which approximates the true filtering distribution. The perturbed observation version of the algorithm is studied, without and with variance inflation. Without variance inflation well-posedness of the filter is established; with variance inflation accuracy of the filter, with respect to the true signal underlying the data, is established. The algorithm is considered in discrete time, and also for a continuous time limit arising when observations are frequent and subject to large noise. The underlying dynamical model, and assumptions about it, is sufficiently general to include the Lorenz '63 and '96 models, together with the incompressible Navier–Stokes equation on a two-dimensional torus. The analysis is limited to the case of complete observation of the signal with additive white noise. Numerical results are presented for the Navier–Stokes equation on a two-dimensional torus for both complete and partial observations of the signal with additive white noise.

Well-posedness of the plasma–vacuum interface problem

Abstract: We consider the free-boundary problem for the plasma–vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma–vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma–vacuum interface problem in suitable anisotropic Sobolev spaces. The proof is based on the results proved in the companion paper (Secchi and Trakhinin 2013 Interfaces Free Boundaries 15 323–57), about the well-posedness of the homogeneous linearized problem and the proof of a basic a priori energy estimate. The proof of the resolution of the nonlinear problem given in the present paper follows from the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash–Moser iteration.

Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows

Abstract: Consider a diffusion-free passive scalar θ being mixed by an incompressible flow u on the torus . Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm is bounded below by an exponential function of time. The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data. We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at least for a significant initial period of time. The main idea behind our proof is to use the recent work of Crippa and De Lellis (2008 J. Reine Angew. Math. 616 15–46) making progress towards the resolution of Bressan's rearrangement cost conjecture.

On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds

Abstract: We consider the Fisher–KPP (for Kolmogorov–Petrovsky–Piskunov) equation with a nonlocal interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions, and prove uniform upper bounds for the solutions of the Cauchy problem, as well as upper and lower bounds on the spreading rate of the solutions with compactly supported initial data.

Dominance of chemotaxis in a chemotaxis–haptotaxis model

Abstract: This paper deals with the Neumann problem for the coupled chemotaxis–haptotaxis model of cancer invasion given byfor x ∈ Ω and t > 0 with, respectively, given nonnegative initial data u0 and w0, where χ, ξ and μ are positive parameters, Ω is a bounded domain in , and n ≥ 1, with smooth boundary.The goal of this work is to identify two parallels between the solution behaviour in () and that in the corresponding two-component chemotaxis system obtained when w ≡ 0:• For the latter, it has been known that for any choice of , solutions are global and remain bounded under the conditionThe first result of this paper says that the above statement remains true for arbitrarily large haptotactic ingredients: If () holds, then for any ξ > 0 and all reasonably smooth w0, () possesses a globally defined solution which is bounded in each of its components.• With regard to the qualitative solution behaviour, this work identifies an explicit smallness condition on w0 which under the assumption () asserts exponential decay of w in the large time limit, whereas both u and v persist in a certain sense.

Travelling waves for the cane toads equation with bounded traits.

Abstract: In this paper, we study propagation in a non-local reaction–diffusion–mutation model describing the invasion of cane toads in Australia (Phillips et al 2006 Nature 439 803). The population of toads is structured by a space variable and a phenotypical trait and the space diffusivity depends on the trait. We use a Schauder topological degree argument for the construction of some travelling wave solutions of the model. The speed c* of the wave is obtained after solving a suitable spectral problem in the trait variable. An eigenvector arising from this eigenvalue problem gives the flavour of the profile at the edge of the front. The major difficulty is to obtain uniform L∞ bounds despite the combination of non-local terms and a heterogeneous diffusivity.

Experimental techniques for turbulent Taylor–Couette flow and Rayleigh–Bénard convection

Abstract: Taylor–Couette (TC) flow and Rayleigh–B´enard (RB) convection are two systems in hydrodynamics, which have been widely used to investigate the primary instabilities, pattern formation, and transitions from laminar to turbulent flow. These two systems are known to have an elegant mathematical similarity. Both TC and RB flows are closed systems, i.e. the total energy dissipation rate exactly follows from the global energy balances. From an experimental point of view, the inherent simple geometry and symmetry in these two systems permits the construction of high precision experimental setups. These systems allowfor quantitative measurements of many different variables, and provide a rich source of data to test theories and numerical simulations. We review the various experimental techniques in these two systems in fully developed turbulent states.

On well-posedness and small data global existence for an interface damped free boundary fluid?structure model

Abstract: We address a fluid?structure system which consists of the incompressible Navier?Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. Given sufficiently small initial data, we prove the global-in-time existence of solutions by establishing a key energy inequality which in addition provides exponential decay of solutions.

Symmetry breaking in a bulk-surface reaction-diffusion model for signalling networks

Abstract: Signalling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction–diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis, we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. On the other hand, the second possibility is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow us to investigate the evolution beyond the regime where the linearization applies.

Finite crystallization in the square lattice

Abstract: This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two- and three-body short-ranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.

On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof

Abstract: We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

Yang–Baxter and reflection maps from vector solitons with a boundary

Abstract: Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schrodinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton–soliton and soliton–boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of the set-theoretical reflection equation are introduced.

Biorthogonal ensembles with two-particle interactions

Abstract: We investigate determinantal point processes on [0, +∞) of the form We prove that the biorthogonal polynomials associated with such models satisfy a recurrence relation and a Christoffel–Darboux formula if $ hetainmathbb Q$ , and that they can be characterized in terms of 1 × 2 vector-valued Riemann–Hilbert problems, which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated with our model if w(λ) = e−nV (λ) in the one-cut case with and without a hard edge

Primitive equations with continuous initial data

Abstract: We address the well-posedness of the primitive equations of the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the equations, thus leading to existence and uniqueness of solutions.

Large-degree asymptotics of rational Painlevé-II functions: Noncritical behaviour

Abstract: This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behaviour. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann–Hilbert representation of the rational Painleve-II functions, and supplies leading-order formulae as well as error estimates.

The KdV equation under periodic boundary conditions and its perturbations

Abstract: In this paper we discuss properties of the Korteweg–de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about the long-time behaviour of solutions for perturbed KdV equations.

Decay of correlation for random intermittent maps

Abstract: We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.

Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation

Abstract: This paper is concerned with the global well-posedness and inviscid limits of several systems of Boussinesq equations with fractional dissipation. Three main results are proven. The first result assesses the global regularity of two systems of equations close to the critical 2D Boussinesq equations. This is achieved by examining their inviscid limits. The second result relates the global regularity of a general system of d-dimensional Boussinesq equations to that of its formal inviscid limit. The third obtains the global existence, uniqueness and inviscid limit of a system of 2D Boussinesq equations with the Yudovich-type initial data.

Spectral analysis of hyperbolic systems with singularities

Abstract: We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

Vortex Filament Equation for a Regular Polygon

Abstract: In this paper, we study the evolution of the localized induction approximation (LIA), also known as vortex lament equation, Xt = Xs^ Xss; for X(s; 0) a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that X(s;t) is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau sum. We also study the fractal behavior of X(0;t), relating it with the socalled Riemann’s non-dierentiable function, that, as proved by S. Jaard, ts with the multifractal model of U. Frisch and G. Parisi, for fully developed turbulence.

Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation

Abstract: We consider a degenerate quasilinear Keller–Segel system of fully-parabolic type involving rotation in the aggregative term, where is a bounded convex domain with smooth boundary. Here S(u, v, x) = (si, j)2×2 is a matrix with . Moreover, for all with nondecreasing on [0, ∞). It is shown that whenever m > 1, for any non-negative initial data, which is sufficiently smooth, the system possesses global and bounded weak solution.

Return times distribution for Markov towers with decay of correlations

Abstract: In this paper we prove two results. First we show that dynamical systems with a -mixing measure have in the limit Poisson distributed return times almost everywhere. We use the Chen–Stein method to also obtain rates of convergence. Our theorem improves on previous results by allowing for infinite partitions and dropping the requirement that the invariant measure has finite entropy with respect to the given partition. As has been shown elsewhere, the limiting distribution at periodic points is not Poissonian (but compound Poissonian). Here we show that for all non-periodic points the return times are in the limit Poisson distributed. In the second part we prove that Lai-Sang Young's Markov towers have Poisson distributed return times if the correlations decay at least polynomially with a power larger than 1.

Quasi-static evolution and congested crowd transport

Abstract: We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with drift, and a congested crowd motion model originally proposed by Maury et al (2010 Math. Models Methods Appl. Sci. 20 1787–821). We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele-Shaw solution as m → ∞ provided the drift potential is strictly subharmonic. Next, using the gradient-flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion as m → ∞. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the Hele-Shaw problem. While proving our main results we also obtain a comparison principle for solutions with the minimizing movement scheme based on the Wasserstein metric, of independent interest.

Information barriers for noisy Lagrangian tracers in filtering random incompressible flows

Abstract: An important practical problem is the recovery of a turbulent velocity field from Lagrangian tracers that move with the fluid flow Here, the filtering skill of L moving Lagrangian tracers in recovering random incompressible flow fields defined through a finite number of random Fourier modes is studied with full mathematical rigour Despite the inherent nonlinearity in measuring noisy Lagrangian tracers, it is shown below that there are exact closed analytic formulas for the optimal filter for the velocity field involving Riccati equations with random coefficients for the covariance matrix This mathematical structure allows a detailed asymptotic analysis of filter performance, both as time goes to infinity and as the number of noisy Lagrangian tracers, L, increases In particular, the asymptotic gain of information from L-tracers grows only like ln L in a precise fashion; ie, an exponential increase in the number of tracers is needed to reduce the uncertainty by a fixed amount; in other words, there is a practical information barrier The proofs proceed through a rigourous mean field approximation of the random Ricatti equation Also, as an intermediate step, geometric ergodicity with respect to the uniform measure on the period domain is proved for any fixed number L of noisy Lagrangian tracers All of the above claims are confirmed by detailed numerical experiments presented here

Short- and long-term forecast for chaotic and random systems (50 years after Lorenz's paper)

Abstract: We briefly review a history of the impact of the famous 1963 paper by E Lorenz on hydrodynamics, physics and mathematics communities on both sides of the iron curtain. This paper was an attempt to apply the ideas and methods of dynamical systems theory to the problem of weather forecast. Its major discovery was the phenomenon of chaos in dissipative dynamical systems which makes such forecasts rather problematic, if at all possible. In this connection we present some recent results which demonstrate that both a short-term and a long-term forecast are actually possible for the most chaotic dynamical (as well as for the most random, like IID and Markov chain) systems. Moreover, there is a sharp transition between the time interval where one may use a short-term forecast and the times where a long-term forecast is applicable. Finally we discuss how these findings could be incorporated into the forecast strategy outlined in the Lorenz's paper.

On the degree distribution of horizontal visibility graphs associated with Markov processes and dynamical systems: diagrammatic and variational approaches

Abstract: Dynamical processes can be transformed into graphs through a family of mappings called visibility algorithms, enabling the possibility of (i) making empirical time series analysis and signal processing and (ii) characterizing classes of dynamical systems and stochastic processes using the tools of graph theory. Recent works show that the degree distribution of these graphs encapsulates much information on the signals' variability, and therefore constitutes a fundamental feature for statistical learning purposes. However, exact solutions for the degree distributions are only known in a few cases, such as for uncorrelated random processes. Here we analytically explore these distributions in a list of situations. We present a diagrammatic formalism which computes for all degrees their corresponding probability as a series expansion in a coupling constant which is the number of hidden variables. We offer a constructive solution for general Markovian stochastic processes and deterministic maps. As case tests we focus on Ornstein–Uhlenbeck processes, fully chaotic and quasiperiodic maps. Whereas only for certain degree probabilities can all diagrams be summed exactly, in the general case we show that the perturbation theory converges. In a second part, we make use of a variational technique to predict the complete degree distribution for special classes of Markovian dynamics with fast-decaying correlations. In every case we compare the theory with numerical experiments.